Space-Filling Curves in Information Visualization Jiwen Huo [email protected].

Space-Filling Curves Space-Filling Curves in Information Visualizationin Information Visualization

Jiwen [email protected]

Space-filling curvesSpace-filling curves

Definition: A continuous curve which passes through

every point of a closed n-cell in Euclidean n-space En is called a space filling curve (SFC).

Application of SFCApplication of SFC

Mapping multi-dimensional space to one dimensional sequence

Applications in computer science: Database multi-attribute access Image compression Information visualization ……

Categories of SFCCategories of SFC

Non-recursive Z-Scan Curve Snake Scan Curve

Recursive Hilbert Curve Peano Curve Gray Code Curve

Non-recursive Space Filling Non-recursive Space Filling CurvesCurves

Z-Scan Curve Snake Scan Curve

Recursive Space Filling CurvesRecursive Space Filling Curves

Hilbert Curve

Gray Code Curve

Peano Curve

3D SFC3D SFC

Z Curve Peano Curve

Hilbert Curve

Properties of SFCsProperties of SFCs

Coherent in Continuity Clustering Property

Direction Preserving

Good

Hilbert CurveZ Curve

Good

Z CurveHilbert Curve

Coherent in ContinuityCoherent in Continuity

Clustering propertyClustering property

Z

H

Direction PreservingDirection Preserving

P1: (X1, Y1) X1 > X2> X3

P2: (X2, Y2) Y1 > Y2> Y3

P3: (X3, Y3)

Z(P1) > Z(P2) > Z(P3)

H(P3) < H(P1) < H(P2)

SFC in Information VisualizationSFC in Information Visualization

Example Data Color (R, G, B: [0, 255]) Data with obvious geometric pattern

4D Hyper Sphere Data without obvious geometric pattern

Iris flowers (5 attributes, 3 classes)

Example SFC Z-Curve Hilbert Curve

Visualizing Color (RGB)Visualizing Color (RGB)

(160, 100)

(160, 100, 100)

100

?

Visualizing Color (RGB)Visualizing Color (RGB)

Z-Curve Mapping

Visualizing Color (RGB)Visualizing Color (RGB)

Hilbert Curve Mapping

Visualizing Color (RGB)Visualizing Color (RGB)

Green & Blue

Red

Z

H

Visualizing 4D Hyper-Sphere SurfaceVisualizing 4D Hyper-Sphere Surface

Z-Curve Hilbert Curve

The Visualization of Iris FlowersThe Visualization of Iris Flowers

Z-Curve Hilbert Curve

Extension to Hilbert CurveExtension to Hilbert Curve

The Visualization of Iris FlowersThe Visualization of Iris Flowers

Extended Hilbert Curve

Questions? Questions?

Suggestions? Suggestions?